Multi-machine Power System Stabilizer Design Using Quantitative Feedback Theory

Power System Stabilizers (PSSs) are used to enhance damping of power system oscillations through excitation control of synchronous generator. The objective of the PSS is to generate a stabilizing signal, which produces a damping torque component on the generator shaft. Conventional PSSs are designed with the phase compensation technique in the frequency domain and include the lead-lag blocks whose parameters are determined according to a linearized power system model. The performance of Conventional PSSs (CPSSs) depends upon the generator operating point and the system parameters, but a reasonable level of robustness can be achieved depending on the tuning method. To overcome the drawbacks of CPSS, numerous techniques have been proposed in literatures. In this paper a robust method based on Quantitative Feedback Theory (QFT) Algorithm is used for tuning the PSS parameters. The proposed QFT-PSS is evaluated at a multi machine electric power system. The simulation results clearly indicate the effectiveness and validity of the proposed method.


INTRODUCTION
Large electric power systems are complex nonlinear systems and often exhibit low frequency electromechanical oscillations due to insufficient damping caused by adverse operating.These oscillations with small magnitude and low frequency often persist for long periods of time and in some cases they even present limitations on power transfer capability (Liu et al., 2005).In analyzing and controlling the power system's stability, two distinct types of system oscillations are recognized.One is associated with generators at a generating station swinging with respect to the rest of the power system.Such oscillations are referred to as "intra-area mode" oscillations.The second type is associated with swinging of many machines in an area of the system against machines in other areas.This is referred to as "inter-area mode" oscillations.Power System Stabilizers (PSS) are used to generate supplementary control signals for the excitation system in order to damp both types of oscillations (Liu et al., 2005).The widely used Conventional Power System Stabilizers (CPSS) are designed using the theory of phase compensation in the frequency domain and are introduced as a lead-lag compensator.The parameters of CPSS are determined based on the linearized model of the power system.Providing good damping over a wide operating range, the CPSS parameters should be fine tuned in response to both types of oscillations.Since power systems are highly nonlinear systems, with configurations and parameters which alter through time, the CPSS design based on the linearized model of the power system cannot guarantee its performance in a practical operating environment.Therefore, an adaptive PSS which considers the nonlinear nature of the plant and adapts to the changes in the environment is required for the power system (Liu et al., 2005).In order to improve the performance of CPSSs, numerous techniques have been proposed for designing them, such as intelligent optimization methods (Linda and Nair, 2010;Yassami et al., 2010;Sumathi et al., 2007;Jiang and Yan, 2008;Sudha et al., 2009) and Fuzzy logic method (Hwanga et al., 2008;Dubey, 2007).Also the application of robust control methods for designing PSS has been presented by Gupta et al. (2005), Mocwane andFolly (2007), Sil et al. (2009) and Bouhamida et al. (2005).
In this study a robust method based on Quantitative Feedback Theory (QFT) Algorithm is used for tuning the PSS parameters.A multi machine electric power system is considered as case study.Simulation results show that the proposed method greatly enhances the dynamic stability of multi machine power system.

SYSTEM UNDER STUDY
In this study IEEE 14 bus test system is considered to evaluate the proposed method.The system data are completely given in IEEE standards.The system uncertainties are obtained by 40% changing load and parameters from their nominal values.Figure 1 shows the proposed test system.(1) where, i = 1, 2, 3, 4, 5 (the generators: 1 to 4); *, rotor angle; T, rotor speed; P m , mechanical input power; P e , electrical output power; E ' q , internal voltage behind x ' d ; E fd , equivalent excitation voltage; Te, electric torque; T ' do , time constant of excitation circuit; K a , regulator gain; T a , regulator time constant; V ref , reference voltage; V t , terminal voltage.
Linear dynamic model of the system: In the PSS design, the power system is usually linearized in order to perform the small signal analysis.Therefore, the system in Eq. ( 1) is linearized around one operating condition of the power system and is given in Eq. ( 2), (2) where, A is the power system state matrix, B is the input matrix, C is the output matrix and D is the feed-forward matrix.From Eq. ( 2), the transfer functions of power system from inputs (such as mechanical input (T m ) or excitation system input) to outputs (such as )T) can be easily obtained.

POWER SYSTEM STABILIZER
As mentioned before, in large interconnected power systems, the damping torque of system is reduced and system need to PSS for stability.The basic function of PSS is to add damping torque to the generator rotor oscillations by controlling its excitation using auxiliary stabilizing signal.To provide damping, the stabilizer must produce a component of electrical torque in phase with the rotor speed deviations (Kundur, 1993).The PSS configuration is given in as (3), where, T is the speed deviation in p.u.This type of PSS consists of a washout filter, a dynamic compensator.The output signal is fed as a supplementary input signal to the excitation of generator.The washout filter, which is a high pass filter, is used to reset the steady state offset in the PSS output.In this paper the value of the time constant (Tw) is fixed to 10 s.The dynamic compensator is made up to two leadlag stages with time constants, T1-T4 and an additional gain K DC .(3) The major point in the PSS design is to find the optimal values of K DC and T1-T4.In this paper a robust method is used to find the best values of the proposed parameters.Where, the optimum values of K DC and the time constants of T1-T4 are obtained by using QFT in the next section a brief introduction about the QFT is presented.

QUANTITATIVE FEEDBACK THEORY
Quantitative Feedback Theory (QFT) is a unified theory that emphasizes to use of feedback for achieving the desired system performance tolerances despite plant uncertainty and plant disturbances.QFT quantitatively formulates these two factors as following form: The object is to guarantee that the control ratio (system transfer function) T R = Y/R is a member of J R and T D = Y/D is a member of J D for all P(S) in D. QFT is essentially a frequency-domain technique and in this paper is used for Multiple Input-Single Output (MISO) systems.It is possible to convert the MIMO system into its equivalent sets of MISO systems to which the QFT design technique is applied.The objective is to solve the MISO problems, i.e., to find compensation functions which guarantee that the performance tolerance of each MISO problem is satisfied for all P in D. The detailed step-by-step procedure to design controllers using QFT technique is given by Dazzo and Houpis (1988) and Horowitz (1982).

DESIGN METHODOLOGY
In this section the PSS parameters tuning based on the QFT is presented.The test system has five generators and it is possible to install five PSSs.In this paper just one PSS is considered to install on generator 1.Based descriptions in previous section, the power system transfer function (plant) and PSS (controller) can be considered like a control loop as shown in Fig. 2. It is clearly seen that the system is a SISO system and compensator (PSS) will be designed based on the plant.The power system transfer function from excitation Fig. 2: The structure of control system Fig. 3: Templates of plant system ()E f ) to speed deviations ()T) can be obtained by using (2).
Based QFT technique (Dazzo and Houpis, 1988;Horowitz, 1982) the first step in the design process is to plot the plant uncertainties in Nichols diagram.This plot is known as system templates.The Templates of plant at various operating conditions are obtained by MATLAB software in some frequencies and shown in Fig. 3.The PSS is a cascade compensator and designed so that the variation of output response ()T) be within the acceptable range under the uncertainties of plant.
In this problem, the output signal )T should driven back to zero after disturbances and in fact the system outputs are regulated by controllers.It means that in this problem the controller has regulatory characteristics and tracking characteristics are not considered.Therefore considering the tracking specifications is not necessary and consequently the tracking bounds are not considered.The output response ()T) is acceptable if its magnitude be below the limits given by the disturbance rejection bounds.The disturbance rejection bounds are obtained according to QFT method using QFT toolbox of MATLB software.Since in this case the tracking bounds have not been considered, so the disturbance rejection bounds (B D (jTi)) are considered as composite bounds (B O (jTi)).Also, minimum damping ratios .for the dominant roots of the closed-loop system is considered as .= 1.2, this amount, on the Nichols chart establishes a region which must not be penetrated by the template of loop shaping (L O ) for all frequencies.The boundary of this region is referred to as U-contour.The U-contour and composite bounds (B O (jTi)) and an optimum loop shaping (L 1 ) based these bounds are shown in Fig. 4. Figure 4 shows   ---------------------------------------------------------------- that the nominal open-loop transfer function (loopshaping) is exactly based QFT bounds and according to QFT theory, the design objectives have been met.The PSS parameters are accuracy calculated using the proposed loop shaping and the results are listed in Table 1.Also in order to show the effectiveness of the QFT method, the parameters of stabilizer are tuned using Genetic Algorithms (GA).In GA case, the performance index is considered as Integral of the Time multiplied Absolute Error (ITAE) which is given as (4). (4) where, the design procedure can be formulated as the following constrained optimization problem, where the constraints are the PSS parameter bounds (Shayeghi et al., 2010) The optimal parameters of stabilizer are obtained by using GA and listed in Table 2.

SIMULATION RESULTS
Simulations are carried out on the test system given in section 2. To evaluate the system performance under different disturbances, two scenarios of fault are considered as follows: Scenario 1: 10 cycle three-phase short circuit in bus 3 Scenario 2: 6 cycle three-phase short circuit in bus 1 The simulation results are presented in Fig. 5-9 GA-PSS (dashed line).The simulation results show that applying the supplementary stabilizer signal greatly enhances the damping of the generator angle oscillations.
The results clearly show that in large electric power systems, PSS can successfully increase damping of power system oscillations.The QFT-PSS has better performance than GA-PSS and results clearly show the validity of QFT method.QFT controller may be used to increase power system operation flexibility and controllability, to enhance system stability and to achieve better utilization of existing power systems.With changing fault, GA-PSS cannot guarantee a robust and acceptable response.But QFT-PSS is designed for a family of plants and with changing system operating condition, this PSS is robust under system changing and system responses are in the range of acceptable.

CONCLUSION
In this study dynamic stability of a multi machine electric power system has been successfully improved by using Power System Stabilizer.The PSS parameters were obtained by using QFT method.The power system has been installed with just one PSS.This application of just one PSS is near to real world applications.The simulation results on a multi machine electric power system showed that PSS can greatly enhance damping of power system oscillations in large electric power systems.Therefore the proposed method is a feasible and appropriate method to enhance dynamic stability of large multi machine electric power systems.The proposed PSS is very feasible and easy to implementation.

Fig. 1 :
Fig. 1: IEEE 14 bus test system Nonlinear dynamic model of the system: The nonlinear dynamic model of the system is given as (1).
acceptable command or tracking input-output relations and sets J D = {T D } of acceptable disturbance input-output relations.C Sets D = {P} of possible plants.

Table 1 :
Parameters of stabilizer using QFT QFT algorithm

Table 2 :
Parameters of stabilizer using GA